=Paper=
{{Paper
|id=Vol-3925/paper11
|storemode=property
|title=Predictive control for failure risk assessment during navigation equipment operation
|pdfUrl=https://ceur-ws.org/Vol-3925/paper11.pdf
|volume=Vol-3925
|authors=Oleksiy Zuiev,Oleksandr Solomentsev,Maksym Zaliskyi,Olga Shevchenko,Alina Osipchuk
|dblpUrl=https://dblp.org/rec/conf/cmigin/ZuievSZSO24
}}
==Predictive control for failure risk assessment during navigation equipment operation==
https://ceur-ws.org/Vol-3925/paper11.pdf
Predictive control for failure risk assessment during
navigation equipment operation
Oleksiy Zuiev1,†, Oleksandr Solomentsev1,†, Maksym Zaliskyi1,∗,†, Olga Shevchenko1,† and
Alina Osipchuk1,†
1
National Aviation University, Liubomyra Huzara Ave. 1, Kyiv, 03058, Ukraine
Abstract
Navigation equipment may fail during operation. As a result, this equipment becomes inoperable and risks
arise related to the integrity of navigation systems and flight safety. To reduce the specified risks and losses
caused by them, it is necessary to solve the problems of increasing the reliability and operational efficiency.
The main methods of solving these problems are the application of various types of redundancy, the
development and implementation of intelligent data processing technologies, the development of decision-
making schemes regarding corrective and preventive actions, the synthesis of methods for prediction of the
technical state of equipment, the development of algorithms and methods for diagnostic, and others.
Prediction is an effective method of reducing failure risks. This approach is based on the determination of
estimates of the future values of the determining parameters of the equipment and the formation of control
influences on the prevention of exceeding the values of these parameters of the given tolerances. During
the synthesis and analysis of prediction algorithms, it is advisable to determine efficiency indicators in the
form of the veracity of predictive control. This paper considers the problem of predictive control procedure
synthesis for decreasing the risks of equipment failure.
Keywords
risks assessment, operation, radio equipment, predictive control, veracity, decision-making 1
1. Introduction
Analysis of theoretical results and practice of operating modern devices of communication,
navigation and surveillance (CNS) indicate the need for wide application of information technologies
for processing operational data regarding the operation of these devices and further modernization
of the operational system (OS) [1, 2]. Control means of modern CNS system obtain a large amount
of data about their technical state (TS), but the algorithms for processing this data are not provided
[3, 4]. This fact leads to the limitation possibilities of operation processes optimization for CNS
devices. In addition, the real operating conditions of specific equipment are not fully taken into
account [5]. That is, information technologies are practically not used to directly analyze the
operation processes, which does not allow purposeful and effective improvement of the OS.
The construction of OS can be based on the implementation of system and process approaches
[6, 7]. These processes must be carried out under managed and controlled conditions, which involve
the performance of operations to regulate the parameters of individual means and components of
their OS [8]. This action strategy is an adaptive approach to the management of operational
processes. To implement adaptive operation, it is advisable to use the following measures:
CH&CMiGIN’24: Third International Conference on Cyber Hygiene & Conflict Management in Global Information Networks,
January 24–27, 2024, Kyiv, Ukraine
∗
Corresponding author.
†
These authors contributed equally.
0801zuiev@gmail.com (O. Zuiev); avsolomentsev@ukr.net (O. Solomentsev); maximus2812@ukr.net (M. Zaliskyi);
olha.shevchenko@npp.nau.edu.ua (O. Shevchenko); alina.osipchuk2012@gmail.com (A. Osipchuk)
0000-0002-4520-3288 (O. Zuiev); 0000-0002-3214-6384 (O. Solomentsev); 0000-0002-1535-4384 (M. Zaliskyi); 0009-0001-
7670-8043 (O. Shevchenko); 0000-0002-9053-2072 (A. Osipchuk)
© 2025 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
1. Identify the main factors that should be taken into account for the implementation of
adaptive operation.
2. Determine the main functions of the OS of the CNS devices.
3. Justify the main actions and operations that must be performed during operation (control,
adjustment, and others).
4. Analyze schemes of information interaction of individual components of OS.
5. Consider possible options for evaluating the effectiveness of the application of adaptive
operation [9, 10].
The operation system includes a set of products, means of operation, performers and
documentation, which establishes the rules of their interaction for the performance of operation
tasks [11]. The main functions of the OS of the CNS devices are:
• organization, coordination and control of the technical operation of the facilities and means
of the CNS;
• carrying out organizational and technical measures regarding modernization, maintenance
of operational readiness and extension of the service life of the CNS devices;
• planning, organization, coordination and control of work on the certification of services and
facilities of the CNS equipment and aviation telecommunications;
• planning, coordination and control of the flight inspections of CNS devices;
• organization and control of work on improving the qualifications of the CNS devices
specialists;
• improvement and introduction of new methods of work organization, including based on the
use of modern information technologies [12, 13].
One of the components of OS is the system of technical maintenance [14]. The maintenance
process of CNS devices belongs to the class of complex systems. The analysis shows that
maintenance processes are characterized by the presence of all the signs of a complex system [15].
Indeed, a large number of different elements interact in it, which have a single functional goal –
radio technical support of the production activity of the aviation enterprise [16]. One of the areas of
optimization of maintenance processes of the CNS devices is the optimization of individual
components of maintenance. The main components of maintenance are:
• control of the TS of the CNS devices;
• regulation of the determining parameters of the CNS devices by the implementation of
controlling influences;
• preventive replacement of blocks, nodes, elements in CNS devices [17, 18].
Because of the maintenance control, such elements of CNS devices are revealed. The condition of
elements can lead to the failure of the equipment as a whole. Adjustment, regulation or replacement
of such elements is carried out, thus it is possible to prevent failures. The regulation consists of the
implementation of controlling influences (CI) on the determining parameters (DP) of the CNS devices
based on the information about their obtained after the implementation of control operations [19].
Controlling influences should be carried out in case of reaching the DP of the limits of safety
tolerances [20]. The purpose of the implementation of the CI is to bring the values of the regulated
DPs, which are controlled, to the nominal values [21, 22].
It is known that long-term statistics, data on the nature of events (planned and sudden shutdowns,
maintenance and repair of equipment, and others), as well as diagnostic data (measurement results
on operating equipment) contain a certain amount of useful information [23, 24]. The analysis of the
specified statistics allows solving a complex of tasks such as estimating the current state of the
equipment and prediction the reliability of the equipment as a whole with the selection and ranking
of negative factors, the elimination of which will lead to its increasing at the given operation interval.
Today it becomes necessary to apply adequate methods of processing statistical series and
building special adaptive algorithms of models for making decisions on operational reliability
management of specific types of the CNS devices. The implementation of modern maintenance
regulations involves the application of the tools of the theory of controlled Markov random processes
in the conditions of stochastic uncertainty of information about the parameters of the CNS devices,
creates the necessary conditions for the development of models for evaluation and prediction the
parameters of operation management [25]. The development of models, algorithms, methods, and
computational procedures that use the statistical data and the results of diagnostic control of
equipment allows performing calculations and prediction of probabilistic characteristics of random
events flows, estimation of the criticality rank of equipment before failures, determination of the
priority order of its prevention and optimization of maintenance parameters and adjustment at the
considered operation interval [26].
In general, the term prediction is understood as a statement that contains an indication of a spatial
or temporal interval of finite size, within which the predicted event will occur [27]. As a result of the
implementation of certain algorithms for processing information about the object of research in the
prediction process, an estimation of certain characteristics of the object at a biased moment in time
is carried out. Data analysis in the case of predicting states or events is more complex than data
analysis from previous experience.
As a result of prediction the technical state of the CNS devices, the value of the set of DPs is
evaluated. At the same time, if each of the parameters determines qualitatively different properties
of the object, then an independent prediction of individual parameters is carried out, the final
decision about the object state is made based on a set of decisions about the state of each of the
parameters [28]. If all parameters qualitatively determine one property of the object, then the vector
characteristic of the suitability of the CNS devices in a certain way turns into a generalized
parameter, based on the results of its control, the technical state of the CNS devices is assessed [29].
The practical application of adaptive algorithms of predictive control involves reducing the errors of
making decisions about the object state in the presence of a priori uncertainty [30, 31]. The veracity
of predictive control is a quantitative measure of the objectivity of decisions made as a result of
prediction [32, 33]. The veracity of control depends not only on the objectivity of the actual
prediction, but also on the effectiveness of the operation systems of CNS devices as a whole [20, 35].
2. Problem statement
Consider the statement of the problem for this research. There is the system that provides some
⃗ can occur related to the systems under its control. The risks can be considered from the
services (for example, radio support of flights). During non-serviceability of this system, the risks
different points of view, including social, economic, political, safety, and others. In general, risks are
estimated in probability terms and for quantitative assessment it is necessary to have information on
costs. In our case, we suppose that risk depends on following parameters:
ℎ ⃗;
• efficiency indicator (for decision-making on TS it can be the veracity) ;
⃗;
• data processing algorithms
the type of prediction ⃗;
• models of DPs and reliability parameters
cost function ⃗ ;
•
⃗.
•
• data and their features of collecting and storage
⃗= ( ! ℎ ⃗, ⃗$ ⃗, ⃗%, ⃗ ).
Therefore, the risk can be determined as a function
The purpose of this paper is to obtain analytical equations for estimating the efficiency of
predictive control procedures for CNS devices.
3. Veracity of predictive control of navigation equipment
Let’s consider the veracity indicators of the predictive control of the equipment operation. While
is made at the prediction interval τ( .
carrying out the predictive control of the operation of navigation equipment, a decision about its TS
Predictive control (PrC) of product operation eliminates a significant drawback of running control
τ) and the time of its use according to the purpose τ) * τ+,- , thereby increasing the effectiveness of
(RC), which consists of the time discrepancy of making a decision about the product serviceability
control and, accordingly, the effectiveness of products use [36, 37]. The initial information for
the limits of permissible values of DP . (⃗ are established in the form of warranty tolerances
decision-making for the PrC is the results of the RC. In the technical documentation for the product,
/Ω12 ; Ω42 5. At the same time, the event 6 (τ( ), which determines the condition of product
6 !τ( %: 8. (⃗ !τ( % ∈ /Ω12 ; Ω42 5:.
serviceability in the prediction interval, is presented in the form:
(1)
Due to the imperfection of the PrC systems, there may be errors when making the decision about
(τ( ), which determines the
the workability of the product, i.e., a serviceable product may be rejected, and an unserviceable
product may be deemed suitable. At the same time, the event
conditions for making a decision about the product serviceability in the prediction interval, will be
!τ( %: 8< (⃗ ∈ /∆12 ; ∆42 5( :.
presented in the following form:
(2)
Graphically, the operation of forming decisions at PrC and taking into account the above
mentioned is shown in Figure 1.
Figure 1: Graph of the forming decisions for PrC.
1) !τ( % = ( 6 !τ( %) is the probability of serviceable state of product at the prediction
Figure 1 contains the following parameters:
interval τ> ;
2) !τ( % = ( 6 !τ( %????????????!τ( %) is the unconditional probability of recognition of a
3) @!τ( % = (????????
6 !τ( % !τ( %) is the probability of recognizing an unsuitable product
suitable product as unsuitable based on the results of the PrC (manufacturer’s risk of PrC);
4) A!τ( % = ( 6 !τ( % !τ( %) is the probability of recognition as suitable based on the
as suitable based on the results of the PrC (customer risk during PrC);
5) B!τ( % = ( 6 !τ( %) is the probability of recognizing the product as suitable based on the
results of the PrC of a suitable product;
results of the PrC;
6) C!τ( % = (????????????!τ( %/ 6 !τ( %) is the conditional probability of recognizing a suitable
product as unsuitable based on the results of the PrC (this is the probability of the first type error
7) E!τ( % = ( !τ( %/????????
6 !τ( %) is the conditional probability of recognizing an
during the PrC);
unsuitable product as suitable based on the results of the PrC (this is the probability of an error of
the second kind during the PrC).
The probabilities of the products control results during the PrC taking into account the accepted
!τ( % = !τ( %C!τ( % = !τ( % − A!τ( %;
designations and in accordance with Figure 1 are in the following equation:
@!τ( % = !τ( %E!τ( % = B!τ( % − A!τ( %;
(3)
B!τ( % = !τ( % − !τ( % * @!τ( %;
(4)
G!τ( % = ? !τ( % − @!τ( %;
(5)
A!τ( % = !τ( % − (H).
(6)
(7)
Let’s consider possible quantitative estimation of decisions veracity during PrC. Since the task of
the PrC is to make the decision about the suitability or unsuitability of products before performing
tasks, during predictive control of serviceable products, there can be two such solutions: the product
is considered suitable (serviceable) or a failure is observed.
Therefore, for the quantitative estimation of the veracity of the decisions made during the PrC, it
is necessary to select the following characteristics, which in general have different numerical values.
We will evaluate quantitatively:
– decision veracity about “suitable” state (a posteriori probability of serviceability in the
P(τ( ) − A(τ( ).
prediction interval of the product determined as “suitable” according to the results of the PrC)
I
!τ( % = ;
P(τ( ) − A(τ( ) * B(τ( )
(8)
– the decision veracity about “unsuitable” state (a posteriori probability of unserviceability in the
1 − P(τ( ) − В(τ( ).
prediction interval of the product determined as “unsuitable” according to the results of the PrC)
M
!τ( % = ;
1 − P(τ( ) − A(τ( ) * B(τ( )
(9)
– absolute veracity (a posteriori probability of making error-free decisions for the prediction
Р
!τ( % = 1 − !τ( % − @!τ( %.
interval according to the results of the PrC)
(10)
determined by the values of the manufacturer’s risk (τ) ) and customer’s risk @(τ) ).
Analysis of equations (8) – (10) shows that the indicators of decisions veracity of the PrC are
Let’s consider mathematical model of the forming decisions for the preventive control of the
determining parameter.
As shown above, the result of the PrC is the decision on the serviceability of the product in
prediction interval. Quantitative results of the product serviceability PrC are initiate data for making
decision. To describe the process forming decisions at PrC, we will use a mathematical model of PrC
[20] that defines PrC as a sequence of performing individual operations with random consequences.
by an a priori random process Q(τ) ), statistically determined on the time interval /0, τ) 5.
Let’s assume that the change over time of the DP of a set of products of the same type is described
One of the implementations of this process, which corresponds to the change in the DP of the
τS τT … τV , τ) : 0 ≤ τS < τT , … , < τV < τ) , which precede the prediction interval τ( .
specific product, is observed by means of running control at discrete moments of time,
of ordinates of its true values at the moments of changes τS , τT … τ) will be presented in the form of
Taking into account the nature discreteness of the moments and DP observations results, the set
Y = ZQ(τS ) ∈ B[ , Q(τT ) ∈ B\ , … , Q(τ) ) ∈ B] }, Y ∈ _,
some trajectory – changes of the true values of the DP [8]
where _ is set of trajectories of change of true DP values in observation intervals; B[ , B\ , … , B] are
(11)
indexes, in which there were true values of DP at the moments of its measurements; ∈ , ` ∈
, … , a ∈ are discrete values of DP at the moments of measurement, is set of parameter values
distinguished by RC means. Each trajectory 1 is characterized by the probability 1 = ( 1 ) of
Set _ is characterized by the matrix. This matrix contains a series of probabilities for the existence
existence.
of true DP trajectories determined by the random process Q(τ), which is an input for the running
control operation:
b 2>
c
b = ‖ S T… Y … 1 ‖, e Y =1.
Y∈1
(12)
The reliability component of the PrC, due to the errors of the RC, will be denoted by the index " "
The set at PrC consists of a subset f of serviceable states and a subset g of non-serviceable
for the considered indicator.
states, f ∈ , g ∈ , (f ∩ g = ∅).
Quantitative results of DP measurements (jS ), (jT ) … (j) ) are transformed by RC means into
the numbers of its various states [ , \ , … ] , ∈ , ` ∈ , … , k ∈ , based on the set
[ , \ , … ] obtained in the observation interval /jS , jT … j) 5. The control system (CS) with the
help of its prediction tool forms the prediction result (j> ) = ( [ , \ , … ] ), which belongs to
many l possible states and represents some characteristic of the posterior random process
Q(j)/ [ , \ , … ] , which describes possible changes in the values of the DP in the prediction
Then the CS determines the membership of the prediction result to the set , ( ⊂ l), that
interval given its initial values.
corresponds to the results “Suitable”. The PrC based on this definition with the correspondingly
chosen decision rule gives a conclusion about the operability of the object instance according to the
We will distinguish between nZ I }; = ???? n, o trajectories formed based on the results of
measured parameter in the prediction interval.
I = Z (jS ) ∈ [ , (jT ) ∈ \ , … , (j) ) ∈ ] }; I ∈ o.
observations by RC means on the variable DP in the observation interval:
Each trajectory I is characterized by the probability of existence p = ( I ). As a result of the
(13)
[ , \ , … > , in the general case, differ from B[ , B\ , … B> , numbers, distinguishable true DP values
error of the selected RC means, obtained at the moment of observation, the numbers of DP states
DP – Y with the probability aY, = Z , / Y } can be perceived as any o-th of the trajectories formed
at the indicated moments of time. In this regard, any -th trajectory of changes in the true values of
based on the results of observations by RC means according to changing the DP.
Conditional probabilities of transitions based on RC results of true trajectories into observed ones
uSS uST ⋯ uS, ⋯ uSI
form a matrix of transition probabilities of RC operations, which is presented in the following form:
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
baqr, b=t t uYS uYT ⋯ uY, ⋯ uYI t t,
Т
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
(14)
u1S u1T ⋯ u1, ⋯ u1I
where: ∑,∈I kYI = 1, = ???? n, _.
The row matrix at the output of the enlarged RC operation is the result of the multiplication of
existence of the set o:
matrices (12) and (14) and contains as its elements the probabilities of the observations trajectories
b Ic b = b 1c b|aqr,
c |
= ‖ S T … , … I ‖,
e = 1, = e
I I Y YI. (15)
,∈I Y∈1
Predictive control of the operations composition differs from RC in the presence of elementary
prediction operations, the resulting joint action of which, as for RC, can be replaced by one enlarged
prediction operation. Consider the probabilistic characteristics of this operation, which differs from
the enlarged RC operation in that the input information for it is not the probabilities of these
operations, but the probabilities of combinations of these probabilities at discrete moments of
observation. Since the used combinations are characterized by the probabilities of the trajectories of
the set _, which are input to the prediction the enlarged operation, then the probabilistic
uSS (j( ) uST (j( ) ⋯ uSz (j( ) ⋯ uS{ (j( )
characteristics of this operation form a transition probability matrix:
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
t u (j ) u (j ) ⋯ u (j ) ⋯ u t
baqr, b = ,{ ( ) ,
(j
(
t t
,S ( ,T ( ,z (
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
(16)
uIS (j( ) uIT (j( ) ⋯ uIz (j( ) ⋯ uI{ (j( )
where ∑z∈{ k,z (j( ) = 1; k,z (j( ) = Z (j( ) ∈ |/ I }, ∈ o is transition probability of transition
o-th trajectory in |-th state in prediction interval.
Multiplying matrices (15) and (16), we obtain the matrix-line of probabilities |B(|, j( )|, | = 1,2
bB(|, j( )b = | ~+q
c |
baqr, b = •B(1, j( )B(2, j( )•.
(
at the output of the enlarged prediction operation:
(17)
Let’s consider the probabilistic characteristics of PrC. For example, the probability of obtaining a
“suitable” result is based on the formula:
B(j( ) = B(1, j( ) = e e Y YI I2 !j( %,
Y∈1 I∈€
(18)
where = Z [ ∈ f, \ ∈ f, … ] ∈ f}.
The considered mathematical model of PrC in the form of a system of matrices describing the
probabilistic characteristics of sequentially performed control operations essentially represents a
Markov model. In this model, the “non-Markovity” of the dependencies of the future values of the
parameter (the predicted value of the parameter in general depends not only on its TS at the present
moment, but also on a number of states at past moments of time) is artificially eliminated by
introducing into the model probabilistic characteristics of the trajectory of values changes controlled
parameter. This introduction made it possible to simplify the calculation of PrC veracity indicators,
and using the generalized mathematical model of object classification [20], to link the characteristics
of RC and PrC.
Let’s consider the characteristics of the decision veracity at the predictive control of the
serviceability of according to the controlled parameter.
parameter, which consists of a subset f of serviceable states and a subset g of non-serviceable states,
During serviceability control, the RC means distinguishes a set of states of the controlled
(f ∩ g = Ø), f * g = . There are such subsets of trajectories , ∈ o that pass only through
subsets of states f of DP. Trajectories are characterized by their own probabilities ‚ = ( ‚ ) and
€ = ( € ) of existence. The change in the parameters of the same type of objects set in the
considered time interval /j( , j) * j+,- ] is described by the random process Q(j), which determines,
in particular, the probability of suitable Р/ (j( )5 and unsuitable Р/????????????(j( )] object
states in the prediction interval j( .
Let’s determine the characteristics of the decision-making veracity for the next decisive rule of
the RC, which is identical to the rule most common in the practice of product serviceability control
Q(j( ) ∈ f, if (j( ) ∈ ƒ(j( ), | = 1 – decision “Suitable”,
[20]:
Q(j( ) ∈„ f, if (j( ) ∈
„ ƒ(j( ), | = 2 – decision “Unsuitable”.
(19)
The most common is the PrC algorithm, in which the product, recognized as unsuitable according
criterion is the fulfillment of the condition I ∈„ f.
to the RC results, is not subject to prediction and is excluded from the RC process. The exclusion
In accordance with the total probability formula [24], we replace the probability (j( ) with the
sum of probabilities Y H(j( ), each of which represents the probability of the coexistence of
…
trajectories and the passage of its continuation in the prediction interval j( through the subset of
the states f :
!j( % = e 1 (j( ),
…
Y∈1
(20)
where 1 !j( % = 8 Y ; ξ(j( ) ∈ f:.
…
? !j( % = e 1‡ (j( ),
Similarly replace
Y∈1
(21)
where 1 !j( % = 8 Y ; ξ(j( ) ∈ g:.
‡
Thus, 1 !j( % * 1 !j( % = 1 .
… ˆ
?????? to
One part characterizes the potential possibility for the continuation of the trajectory ТY , = 1,
With this replacement, the probability of existence of each true trajectory is divided into 2 parts.
result, two new matrices can be formed, the elements of which are functions of the argument j( :
end up in a serviceable state, the second – in a non-serviceable state in the prediction interval. As a
b 1 !j( %b = • S !j( % T !j( % … Y !j( % … 1 !j( %•;
… … … … …
b 1 !j( %b = • S !j( % T !j( % … Y !j( % … 1 !j( %•.
‡ ‡ ‡ ‡ ‡
(22)
Some -th trajectory, Y ∈ f, to which the probabilities 1 !j( % correspond, due to the errors of
…
(23)
the RC operation, may be mistakenly accepted by the RC product as the o-th trajectory of the
observation results, and will be excluded from RC. This leads to the manufacturer’s risk А(Т) (j( ) at
„ ; Q(j( ) ∈ f} =
PrC caused by RC errors
А(Т) (j( ) = Z I∈ e Y, ;
…
Y
I∈I/€
(24)
„ /Q(j( ) ∈ f} = А(Т) (j( )/ (j( ) = А(Т) (j( )/ e Y (j( ) .
C (Т) (j( ) = Z I ∈
…
Y∈1
(25)
Y !j( % corresponds, can be
‡
On the other hand, -th trajectory, Y , to which the probability
mistakenly accepted by the CC product as the o-th trajectory, I , which leads to the customer’s risk
@(j( ) at PrC due to RC errors
@(Т) (j( ) = Z I ∈ ; Q(j( ) ∈ g} = e e Y (j( ) Y, ;
…
Y∈1 ,∈€
(26)
E (Т) (j( ) = Z I ∈ /Q(j( ) ∈ g} = @(Т) (j( )/ ? (j( ) = @(Т) (j( )/ e Y j( ).
‡
Y∈1
(27)
Similarly, the components of the probabilities A (Т) (j( ), G (Т) !j( % are found, which characterize
DP, respectively, in the prediction interval j(
the part of correct decisions in the PrC of serviceable and no serviceable products according to the
A (Т) (j( ) = Z I ∈ ; Q(j( ) ∈ f} = e e Y (j( ) Y, ;
…
Y∈1 ,∈€
(28)
„ ; Q!j( % ∈ g: = e e Y j( ) Y, .
G (Т) !j( % = 8 I ∈
‡
Y∈1 ,∈I
(29)
On the one hand, some -th trajectory, Y ∈ f, to which the probability Y (j( ) corresponds,
…
accepted by the RС as o-th trajectory of the results of observations, I ∈ , may be falsely rejected
due to errors of the prediction operation. This leads to the manufacturer’s risk ( !j( %, due to the
!j( % ∈ | = 2;
errors of the prediction operation at RC. By analogy with (24), we have:
!j( % = Š ‹ = e e e Y !j( % YI ,z !j( %.
( ‡
I ∈ ; Q!j % ∈ f
( Y∈1 ,∈€ zŒ{
(30)
•
On the other hand, some -th trajectory, Y ∈ , to which the probability Y (j( ) corresponds,
‡
accepted by the RC product as the S-th trajectory of the observation results, I ∈ , due to the errors
of the prediction operation, can be rejected with probability G(j( ). By analogy with (29), we have:
!j( % ∈ | = 1;
Ž( !j( % = Š ‹ = e e e Y !j( % YI ,z !j( %.
ˆ
I ∈ ; Q!j % ∈ g
( Y∈1 ,∈€ zŒ{
(31)
•
Graphically, the operation of forming a decision at PrC in accordance with [20], Figure 1 and
taking into account the above mentioned is presented in the form of the probability graph in Figure 2.
1) C (() (j( ) = ( (j( ) ∈„ f / I ∈ ) is the conditional probability of recognizing a suitable
The graph uses the following notations:
product as unsuitable based on the results of the PrC, due to the errors of the prediction operation -
2) E (() (j( ) = ( (j( ) ∈ f / I ∈ „ ) is the conditional probability of recognizing an unsuitable
the probability of an error of the first kind, due to the errors of the prediction operation.
product as suitable based on the results of the PrC, due to the errors of the prediction operation –
the probability of an error of the second kind, due to the errors of the prediction operation.
Figure 2: Graph of decision-making during RC taking into account the possible probabilities of
making erroneous decisions.
Characteristics of the solutions veracity at the RC of the serviceability products, taking into
!j( % = c (j) ) * ( !j( %;
account the accepted designations and in correlation with Figure 2 are in the following relationship:
@!j( % = @c (j) ) − G ( (j) );
A!j( % = A c (j) ) − ( !j( %;
(32)
G!j( % = G c (j) ) * G ( !j( %.
1
Substituting (20), (21), (24), (26), (28), (29) into formula (32) we get:
!j( % = e e Y !j( % YI * e e e Y !j( % Y, ,z !j( %;
… …
YŒS ,∈• Y∈1 ,∈€ zŒ{
(33)
•
@!j( % = e e e Y !j( % Y, ,z !j( %;
‡
Y∈1 ,∈€ zŒ•
(34)
1
C!j( % = e e e Y !j( % YI ,z !j( % ;
…
(35)
YŒS ,∈€ z∈••
1 1
G!j( % = e e Y !j( % YI * e e e Y !j( % Y, ,z !j( %.
‡ ‡
YŒS ,∈ I YŒS ,∈€ zŒ{
(36)
€ •
to subtraction a part that equal to G(j( ) from the risk component @c (j) ), the addition of a part, that
Figure 2 and formula (32) shows that the effect of RC operations with this RC algorithm affects
equal to ( (j( ) to the manufacturer’s risk component c (j) ). As a result, the total risk of the
manufacturer (j( ) increases, and the risk of the customer @(j( ) decreases in comparison with
similar components caused by errors of RC operations.
4. Conclusions
While operating navigation equipment we can observe failures. This causes the equipment
inoperable and gives rise to risks concerning the integrity of navigation systems and flight safety.
To mitigate these risks and minimize associated costs, it is essential to address issues related to
increasing reliability and operational efficiency. Key methods for resolving these challenges include
implementing various forms of redundancy, developing and applying intelligent data processing
technologies, establishing decision-making frameworks for corrective and preventive actions,
synthesizing techniques for prediction of equipment’s technical state, and creating algorithms and
methods for diagnostics, and others. Prediction emerges as the effective strategy for reducing the
risks of equipment failure. This approach relies on estimating future values of DPs and implementing
control measures to prevent these values from exceeding specified tolerances. When synthesizing
and analyzing prediction algorithms, it is advisable to determine efficiency indicators, such as the
veracity of predictive control.
The paper is devoted to the study of issues related to the features of the application of predictive
control and estimation of the veracity of decision-making as a result of prediction. At the same time,
mathematical equations are given that characterize the process of classifying the technical state
during prediction and further estimation of the veracity of decision-making.
The results of the research can be used for the development and modernization of systems for the
operation of navigation devices.
Declaration on Generative AI
The author(s) have not employed any Generative AI tools.
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